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On the Diophantine Equation Gn(x) = Gm(y) with Q (x, y)=0

  • Clemens Fuchs
  • Attila Pethő
  • Robert F. Tichy
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

Let K denote an algebraically closed field of characteristic 0, and let A0,..., Ad–1, G0,..., G d- 1 ∈ K[X] and \( \left( {Gn\left( X \right)} \right)_{n = 0}^\infty \) be a sequence of polynomials defined by the d- th order linear recurring relation
$$ G_{n + d} \left( X \right) = A_{d - 1} \left( X \right)G_{n + d - 1} \left( X \right) + \cdots + A_0 \left( X \right)G_n \left( X \right), for n \geqslant 0. $$
(1)
Furthermore, let P(X) ∈ K[X], deg P ≥ 1. Recently, we investigated the question, what can be said about the number of solutions of the Diophantine equation
$$ Gn\left( X \right) = Gm\left( {P\left( X \right)} \right). $$
(2)

Keywords

Diophantine equations linear recurring sequences S-unit equations 

2000 Mathematics subject classification

Primary 11D45 Secondary 11D04 11D61 11B37 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Clemens Fuchs
    • 1
  • Attila Pethő
    • 2
    • 3
  • Robert F. Tichy
    • 4
  1. 1.Departement MathematikETH ZürichZürichSwitzerland
  2. 2.Faculty of InformaticsUniversity of DebrecenDebrecenHungary
  3. 3.Number Theory Research GroupHungarian Academy of SciencesDebrecenHungary
  4. 4.Institut für Analysis und Computational Number TheoryGrazAustria

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